Convergence of a variational iterative algorithm for nonlocal vibrations analysis of a nanotube conveying fluid
Abstract
The amplitudes of the forced oscillations of a nano-structure conveying fluid are the solutions of an inhomogeneous integral-differential system. This is solved by an easily accessible scheme based on the variational iteration method (VIM), Galerkin’s method and the Laplace transform techniques. The presented method is accompanied by the study of the convergence of the iterative process and of the errors. In the literature, the dynamic response of a viscoelastic nanotube conveying fluid is frequently obtained by an iterative method. This leads to the double convolution products, whose presence will be avoided in the new method proposed in this paper. Thus, the numerical results will be obtained much faster and more accurately.
Keyword : nanobeam conveying fluid, nonlocal calculus, Galerkin’s method, variational iteration method, Laplace transform
How to Cite
Martin, O. (2023). Convergence of a variational iterative algorithm for nonlocal vibrations analysis of a nanotube conveying fluid. Mathematical Modelling and Analysis, 28(3), 360–373. https://doi.org/10.3846/mma.2023.16620
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
R. Ansari, J. Torabi and M.F. Shojaei. An efficient numerical method for analyzing the thermal effects on the vibration of embedded single-walled carbon nanotubes based on the nonlocal shell model. Mechanics of Advanced Materials and Structures, 25(6):500–511, 2018. https://doi.org/10.1080/15376494.2017.1285457
B. Arash and Q. Wang. A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Computational Materials Science, 51(1):303–313, 2012. https://doi.org/10.1016/j.commatsci.2011.07.040
A.R. Askarian, M.R. Permoon and M. Shakouri. Vibration analysis of pipes conveying fluid resting on a fractional Kelvin-Voigt viscoelastic foundation with general boundary conditions. International Journal of Mechanical Sciences, 179:105702, 2020. https://doi.org/10.1016/j.ijmecsci.2020.105702
A.R. Askarian, M.R. Permoon, M. Zahedi and M. Shakouri. Stability analysis of viscoelastic pipes conveying fluid with different boundary conditions described by fractional Zener model. Applied Mathematical Modelling, 103:750–763, 2022. https://doi.org/10.1016/j.apm.2021.11.013
L. Behera and S. Chakraverty. Recent researches on nonlocal elasticity theory in the vibration of carbon nanotubes using beam models. Archives of Computational Methods in Engineering, 24(3):481–494, 2017. https://doi.org/10.1007/s11831-016-9179-y
M. Cajić, D. Karličić and M. Lazarević. Nonlocal vibration of a fractional order viscoelastic nanobeam with attached nanoparticle. Theoretical and Applied Mechanics, 42(3):167–190, 2015. https://doi.org/10.2298/TAM1503167C
A.C. Eringen. Nonlocal Continuum Field Theories. Springer-Verlag, New York, USA, 2002.
S.A. Khuri and A. Sayfy. A Laplace variational iteration strategy for the solution of differential equations. Applied Mathematics Letters, 25(12):2288–2305, 2012. https://doi.org/10.1016/j.aml.2012.06.020
Y. Lei, S. Adhikari and M.I. Friswell. Vibration of nonlocal Kelvin-Voight viscoelastic damped Timoshenko beams. International Journal of Engineering Science, 66-67:1–13, 2013. https://doi.org/10.1016/j.ijengsci.2013.02.004
F. Mainardi. Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London, 2010. https://doi.org/10.1142/p614
F. Mainardi and G. Spada. Creep, relaxation and viscosity properties for basic fractional models in rheology. The European Physical Journal Special Topics, 193:133–160, 2011. https://doi.org/d10.1140/epjst/e2011-01387-1
I. Maron and B. Demidovich. Numerical Calculation Elements. Edition MIR, Moscow, 1973.
O. Martin. Nonlinear dynamic analysis of viscoelastic beams using a fractional rheological model. Applied Mathematical Modelling, 43:351–359, 2017. https://doi.org/10.1016/j.apm.2016.11.033
O. Martin. An iterative algorithm for studying forced vibrations of a nanotube conveying fluid. Mechanics of Advanced Materials and Structures, 29(25):4180– 4192, 2022. https://doi.org/10.1080/15376494.2021.1921317
J. Peddieson, G.R. Buchanan and R.P. McNitt. Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 41(3):305–312, 2003. https://doi.org/10.1016/S0020-7225(02)00210-0
J.N. Reddy and S.D. Pang. Nonlocal continuum theories of beams for the analysis of carbon nanotubes. Journal of Applied Physics, 103(2):23511–23527, 2008. https://doi.org/10.1063/1.2833431
S. Shaw, M.K. Warby and J.R. Whiteman. A comparison of hereditary integral and internal variable approaches to numerical linear solid viscoelasticity. In Proceedings of the Fourteenth Polish Conference on Computer Methods in Mechanics, Poznan, 1997, pp. 183–200, 1997.
Y.Q. Wang, H.H. Li, Y.F. Zhang and J.W. Zu. A nonlinear surfacestress-dependent model for vibration analysis of cylindrical nanoscale shells conveying fluid. Applied Mathematical Modelling, 64(12):55–70, 2018. https://doi.org/10.1016/j.apm.2018.07.016
Y.Q. Wang, Y.H. Wan and J.W. Zu. Nonlinear dynamic characteristics of functionally graded sandwich thin nanoshells conveying fluid incorporating surface stress influence. Thin-Walled Structures, 135(2):537–547, 2019. https://doi.org/10.1016/j.tws.2018.11.023
B. Arash and Q. Wang. A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Computational Materials Science, 51(1):303–313, 2012. https://doi.org/10.1016/j.commatsci.2011.07.040
A.R. Askarian, M.R. Permoon and M. Shakouri. Vibration analysis of pipes conveying fluid resting on a fractional Kelvin-Voigt viscoelastic foundation with general boundary conditions. International Journal of Mechanical Sciences, 179:105702, 2020. https://doi.org/10.1016/j.ijmecsci.2020.105702
A.R. Askarian, M.R. Permoon, M. Zahedi and M. Shakouri. Stability analysis of viscoelastic pipes conveying fluid with different boundary conditions described by fractional Zener model. Applied Mathematical Modelling, 103:750–763, 2022. https://doi.org/10.1016/j.apm.2021.11.013
L. Behera and S. Chakraverty. Recent researches on nonlocal elasticity theory in the vibration of carbon nanotubes using beam models. Archives of Computational Methods in Engineering, 24(3):481–494, 2017. https://doi.org/10.1007/s11831-016-9179-y
M. Cajić, D. Karličić and M. Lazarević. Nonlocal vibration of a fractional order viscoelastic nanobeam with attached nanoparticle. Theoretical and Applied Mechanics, 42(3):167–190, 2015. https://doi.org/10.2298/TAM1503167C
A.C. Eringen. Nonlocal Continuum Field Theories. Springer-Verlag, New York, USA, 2002.
S.A. Khuri and A. Sayfy. A Laplace variational iteration strategy for the solution of differential equations. Applied Mathematics Letters, 25(12):2288–2305, 2012. https://doi.org/10.1016/j.aml.2012.06.020
Y. Lei, S. Adhikari and M.I. Friswell. Vibration of nonlocal Kelvin-Voight viscoelastic damped Timoshenko beams. International Journal of Engineering Science, 66-67:1–13, 2013. https://doi.org/10.1016/j.ijengsci.2013.02.004
F. Mainardi. Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London, 2010. https://doi.org/10.1142/p614
F. Mainardi and G. Spada. Creep, relaxation and viscosity properties for basic fractional models in rheology. The European Physical Journal Special Topics, 193:133–160, 2011. https://doi.org/d10.1140/epjst/e2011-01387-1
I. Maron and B. Demidovich. Numerical Calculation Elements. Edition MIR, Moscow, 1973.
O. Martin. Nonlinear dynamic analysis of viscoelastic beams using a fractional rheological model. Applied Mathematical Modelling, 43:351–359, 2017. https://doi.org/10.1016/j.apm.2016.11.033
O. Martin. An iterative algorithm for studying forced vibrations of a nanotube conveying fluid. Mechanics of Advanced Materials and Structures, 29(25):4180– 4192, 2022. https://doi.org/10.1080/15376494.2021.1921317
J. Peddieson, G.R. Buchanan and R.P. McNitt. Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 41(3):305–312, 2003. https://doi.org/10.1016/S0020-7225(02)00210-0
J.N. Reddy and S.D. Pang. Nonlocal continuum theories of beams for the analysis of carbon nanotubes. Journal of Applied Physics, 103(2):23511–23527, 2008. https://doi.org/10.1063/1.2833431
S. Shaw, M.K. Warby and J.R. Whiteman. A comparison of hereditary integral and internal variable approaches to numerical linear solid viscoelasticity. In Proceedings of the Fourteenth Polish Conference on Computer Methods in Mechanics, Poznan, 1997, pp. 183–200, 1997.
Y.Q. Wang, H.H. Li, Y.F. Zhang and J.W. Zu. A nonlinear surfacestress-dependent model for vibration analysis of cylindrical nanoscale shells conveying fluid. Applied Mathematical Modelling, 64(12):55–70, 2018. https://doi.org/10.1016/j.apm.2018.07.016
Y.Q. Wang, Y.H. Wan and J.W. Zu. Nonlinear dynamic characteristics of functionally graded sandwich thin nanoshells conveying fluid incorporating surface stress influence. Thin-Walled Structures, 135(2):537–547, 2019. https://doi.org/10.1016/j.tws.2018.11.023